Universal Operator Envelopes and Independent Arithmetic Defects of Additive Ternary Gamma-Rings

Abstract

For an additive ternary $Γ$-ring $T$, we construct the associative rng generated by the two-element, two-index operators on ternary modules and compare it with its realization on the regular module. For a finite-free two-step system $T_μ=E\oplus W$, two integral matrices control complementary parts of this comparison: the middle-relation matrix $B_μ$ presents the hidden quotient occurring in the integral regular kernel, while the visible flattening $Θ_μ$ controls the visible image, its saturation and its nonflat base-change correction. The torsion cokernels of these matrices define an ordered arithmetic defect pair; this pair records selected arithmetic data and is not asserted to be the whole regular kernel. The principal result is a constructive realization theorem: for every ordered pair $(G,H)$ of finite abelian groups, there is a torsion-free two-step ternary $\mathbb Z$-ring whose visible-saturation torsion is $G$ and whose middle-relation torsion is $H$. Thus the two torsion mechanisms, including their prime supports and invariant factors, can be prescribed independently. The proof builds two rank-$(2,1)$ atomic tensor families with complementary Smith profiles and establishes an orthogonal-sum theorem for their presentation matrices. We also identify the standard base-change obstruction attached to these two matrices, describe their determinantal loci, realize classical rank-one matrix-factorization periodicity in an example not induced by an associative $\mathbb Z$-bilinear binary product on its underlying group, and classify all $256$ integral $2\times2\times2$ sign tensors by the resulting defect data. All finite calculations use exact integer arithmetic and are independently reproducible.